Norfolk people use 'Lynn' to refer to King's Lynn. The Oxford Names Companion only recognises King's Lynn and neighbouring West Lynn as valid place names incorporating 'Lynn' in Norfolk. As Mamya says, it's derived from the Celtic for 'the pool'.
I know Great Yarmouth quite well and I've never heard of a district there called 'Lynn'. However, as well as being the name of a school, 'Lynn Grove' is the name of a road in Gorleston, which now effectively forms a suburb of Great Yarmouth. (The school probably just takes its name from the road that it's on).